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New coprime vertex labelings

Dana Ernst
January 30, 2016

New coprime vertex labelings

A coprime vertex labeling is an injective assignment of the labels {1, 2, . . . , n} to the vertices of an n-vertex simple connected graph such that adjacent vertices receive relatively prime labels. I will present new labelings for several infinite families of graphs. No prior knowledge of graph theory will be assumed. Joint work with Nathan Diefenderfer, Michael Hastings, Levi Heath, Briahna Preston, Emily White, and Alyssa Whittemore.

This talk was given by my undergraduate research student Hannah Prawzinsky at the 2016 Nebraska Conference for Undergraduate Women in Mathematics at the University of Nebraska-Lincoln.

This research was supported by the National Science Foundation grant #DMS-1148695 through the Center for Undergraduate Research (CURM).

Dana Ernst

January 30, 2016
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  1. New Coprime Vertex Labelings
    Hannah Prawzinsky
    Joint work with: Nathan Diefenderfer, Michael Hastings, Levi
    Heath, Briahna Preston, Emily White & Alyssa Whittemore
    NCUWM
    January 30, 2016

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  2. What is a graph?
    Definition
    A graph G(V, E) is a set V of vertices and a set E of edges
    connecting some (possibly empty) subset of those vertices.

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  3. Simple graphs
    Definition
    A simple graph is a graph that contains neither “loops” nor
    multiple edges between vertices.
    For the remainder of the presentation, all graphs are assumed
    to be simple. Here is a graph that is NOT simple.

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  4. Connected and unicyclic graphs
    Definition
    A connected graph is a graph in which there exists a “path”
    between every pair of vertices.
    For the remainder of the presentation, all graphs are assumed
    to be connected.
    Definition
    A unicyclic graph is a simple graph containing exactly one
    cycle.
    Here is a unicyclic graph that is NOT connected.

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  5. Infinite families of graphs
    P8
    C12
    S5

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  6. Graph labelings
    Definition
    A graph labeling is an “assignment” of integers (possibly
    satisfying some conditions) to the vertices, edges, or both.
    Formal graph labelings are functions.
    2 3 2 3
    1 4 1 4
    1
    2
    3
    4
    1
    2 3
    4

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  7. Prime vertex labelings
    Definition
    An n-vertex graph has a prime vertex labeling if its vertices are
    labeled with the integers 1, 2, 3, . . . , n such that no label is
    repeated and all adjacent vertices (i.e., vertices that share an
    edge) have labels that are relatively prime.
    1
    6
    7
    4
    9
    2
    3
    10
    11
    12
    5
    8
    Some useful number theory facts:
    • All pairs of consecutive integers
    are relatively prime.
    • Consecutive odd integers are
    relatively prime.
    • A common divisor of two integers
    is also a divisor of their difference.
    • The integer 1 is relatively prime to
    all integers.

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  8. Original motivation for research
    Conjecture (Seoud and Youssef, 1999)
    All unicyclic graphs have a prime vertex labeling.
    Though our research lead to many new results for unicyclic
    graphs, some of which were presented last year, this talk will
    primarily focus on a a specific family of non-unicyclic graphs.

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  9. Known prime labelings
    1 2 3 4 5 6 7 8
    P8
    1
    12
    11
    10
    9
    8
    7
    6
    5
    4
    3
    2
    C12
    1
    2
    6
    5
    4 3
    S5

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  10. Cycle Chains
    Definition
    A cycle chain, denoted Cm
    n
    , is a graph that consists of m
    different n-cycles adjoined by a single vertex on each cycle
    (each cycle shares a vertex with its adjacent cycle(s)).
    Here we show labelings for Cm
    4
    , Cm
    6
    , and Cm
    8
    . The labelings for
    these three infinite families of graphs all employ similar
    strategies.

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  11. Example of C4
    8

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  12. Cycle chain results
    Theorem
    All Cm
    8
    , Cm
    6
    , Cm
    4
    have prime vertex labelings.

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  13. Labeled C5
    8
    1
    2
    3
    4
    5
    6
    7
    8
    15
    11
    10
    9
    1
    12
    13
    14
    19
    18
    17
    16
    15
    22
    21
    20
    29
    25
    24
    23
    19
    26
    27
    28
    33
    32
    31
    30
    29
    36
    35
    34

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  14. Labeled C5
    6
    1
    2
    3
    4
    5 6
    11
    8
    7
    1
    9 10
    16
    13
    12
    11
    14 15
    19
    18
    17
    16
    21 20
    26
    23
    22
    19
    24 25

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  15. Labeled C4
    4
    5
    4
    3
    2
    7
    6
    5
    8
    11
    9
    7
    10
    13
    12
    11
    1

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  16. Labeled C5
    4
    5
    4
    3
    2
    7
    6
    5
    8
    11
    9
    7
    10
    13
    12
    11
    14
    1
    15
    13
    16

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  17. Mersenne Primes
    Definition
    A Mersenne prime is a prime number of the form Mn = 2n − 1.
    There are 48 known Mersenne primes. The first few Mersenne
    primes are:
    M2 = 22 − 1 = 3
    M3 = 23 − 1 = 7
    M5 = 25 − 1 = 31

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  18. Mersenne cycle chains
    Theorem
    All Cm
    n
    , where n = 2k and 2k − 1 is a Mersenne prime, have
    prime labelings.

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  19. Fibonacci Chains
    Fibonacci sequence
    The sequence, {Fn}, of Fibonacci numbers is defined by the
    recurrence relation Fn = Fn−1 + Fn−2
    , where F1 = 1 and F2 = 1.
    1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, . . .
    Proposition
    Any two consecutive Fibonacci numbers in the Fibonacci
    sequence are relatively prime.
    Theorem
    Fibonacci Chains, denoted Cn
    F
    , are prime for all n ∈ N where n is
    the number of cycles that make up the Fibonacci chain.

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  20. Fibonacci Chains (C5
    F
    )
    1
    2
    4
    3
    5
    6
    7 10
    9
    8
    12
    11
    13
    14
    15
    16
    17
    18
    19
    20
    21

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  21. Future Work and Acknowledgements
    Future Work
    • Seoud and Youssef’s Conjecture
    Acknowledgments
    • NCUWM Organizers
    • University of Nebraska—Lincoln
    • Center for Undergraduate Research in Mathematics
    • Northern Arizona University
    • Research Advisors Dana Ernst and Jeff Rushall

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  22. A Labeled 7-Hairy Cycle
    1
    2
    3
    4 5 6
    7
    8
    19
    17
    18
    20
    21
    22
    23
    24
    11
    9
    10
    12
    13
    14
    15
    16
    29
    25
    26
    27
    28
    30
    31
    32

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  23. Prime Vertex Labeling of C5
    P2
    S6
    1
    5
    2
    3 4 6 7
    8
    9
    13
    10
    11
    12
    14
    15
    16
    17
    19
    18
    20
    21
    22
    23
    24
    25
    29
    26
    27
    28
    30
    31
    32
    33
    37
    34
    35
    36
    38
    39
    40

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  24. Prime Vertex Labeling of Cn
    P2
    S3
    S3
    1
    2
    5
    9
    11
    3 4
    6
    7
    8
    10
    12
    13
    14
    15
    16
    19
    23
    25
    17
    18
    20
    21
    22
    24
    26
    27
    28
    29
    32
    31
    35
    41
    30
    33
    34
    36
    37
    38
    39
    40
    42
    43
    44
    47
    51
    53
    45
    46
    48
    49
    50
    52
    54
    55 56

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  25. A Labeled Bertrand Weed Graph
    1
    2
    13
    10
    9
    11
    14
    7
    12
    8
    5
    4
    3
    6

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  26. Example of H8
    (Hastings Helms)
    5
    4
    3
    2
    1
    16
    7
    6
    12
    11
    10
    9
    8
    15
    14
    13

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  27. Book Generalizations
    Here is an example of the prime labeling for Sn × P6
    , in
    particular, S4 × P6
    :
    7 8 9 10 11 12
    13 14 15 16 17 18
    19 20 21 22 5 24
    25 26 27 28 29 30
    6 1 2 3 4 23

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